Conjectured Sprague–Grundy values for king moves on generalized staircases with odd equal parameters

Let $\mathbb{SG}_{\mathrm{King}}(\mathcal{S}_{r,r}^{k})$ denote the Sprague–Grundy value of the impartial chess game with the king move rule on the generalized staircase determined by parameters $r$, $r$, and $k$. Let $r$ be odd and $k\geq 1$. The generalized-staircase king conjecture.

$$\mathbb{SG}_{\mathrm{King}}(\mathcal{S}_{r,r}^{k})=\begin{cases} 0 & \text{if } k\bmod(r+2)\text{ is odd},\\ 1 & \text{if } k\bmod(r+2)\text{ is even and not }0,\\ 2 & \text{if } k\bmod(r+2)=0.\end{cases}$$

This is posed as an open question about computing Sprague–Grundy values for generalized staircases in impartial chess; the supplied text gives no resolution of this case.